Find the values of the following allied angles :
(vii) `cos 135^(@)`

Answer

To find the value of \( \cos 135^\circ \), we can follow these steps:

### Step 1: Identify the angle's quadrant
135° lies in the second quadrant of the unit circle, where cosine values are negative.

### Step 2: Use the reference angle
The reference angle for 135° is calculated as:
\[ 180^\circ - 135^\circ = 45^\circ \]

### Step 3: Apply the cosine function
In the second quadrant, the cosine of an angle is negative. Therefore, we can express \( \cos 135^\circ \) in terms of its reference angle:
\[ \cos 135^\circ = -\cos 45^\circ \]

### Step 4: Find the cosine of the reference angle
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 5: Substitute the value
Substituting the value of \( \cos 45^\circ \) into the equation:
\[ \cos 135^\circ = -\frac{1}{\sqrt{2}} \]

### Step 6: Simplify the expression
We can also express this in a more standard form:
\[ \cos 135^\circ = -\frac{\sqrt{2}}{2} \]

### Final Answer
Thus, the value of \( \cos 135^\circ \) is:
\[ \cos 135^\circ = -\frac{\sqrt{2}}{2} \]

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